Problem: $ F = \left[\begin{array}{rr}-2 & 0 \\ 1 & 1 \\ 4 & -2\end{array}\right]$ $ w = \left[\begin{array}{r}3 \\ 5\end{array}\right]$ What is $ F w$ ?
Answer: Because $ F$ has dimensions $(3\times2)$ and $ w$ has dimensions $(2\times1)$ , the answer matrix will have dimensions $(3\times1)$ $ F w = \left[\begin{array}{rr}{-2} & {0} \\ {1} & {1} \\ \color{gray}{4} & \color{gray}{-2}\end{array}\right] \left[\begin{array}{r}{3} \\ {5}\end{array}\right] = \left[\begin{array}{r}? \\ ? \\ ?\end{array}\right] $ To find the element at any row $i$ , column $j$ of the answer matrix, multiply the elements in row $i$ of the first matrix, $ F$ , with the corresponding elements in column $j$ of the second matrix, $ w$ , and add the products together. So, to find the element at row 1, column 1 of the answer matrix, multiply the first element in ${\text{row }1}$ of $ F$ with the first element in ${\text{column }1}$ of $ w$ , then multiply the second element in ${\text{row }1}$ of $ F$ with the second element in ${\text{column }1}$ of $ w$ , and so on. Add the products together. $ \left[\begin{array}{r}{-2}\cdot{3}+{0}\cdot{5} \\ ? \\ ?\end{array}\right] $ Likewise, to find the element at row 2, column 1 of the answer matrix, multiply the elements in ${\text{row }2}$ of $ F$ with the corresponding elements in ${\text{column }1}$ of $ w$ and add the products together. $ \left[\begin{array}{r}{-2}\cdot{3}+{0}\cdot{5} \\ {1}\cdot{3}+{1}\cdot{5} \\ ?\end{array}\right] $ Fill out the rest: $ \left[\begin{array}{r}{-2}\cdot{3}+{0}\cdot{5} \\ {1}\cdot{3}+{1}\cdot{5} \\ \color{gray}{4}\cdot{3}+\color{gray}{-2}\cdot{5}\end{array}\right] $ After simplifying, we end up with: $ \left[\begin{array}{r}-6 \\ 8 \\ 2\end{array}\right] $